Today we consider subobjects and quotient objects in the category of representations of an algebra . Since the objects are representations we call these “subrepresentations” and “quotient representations”.
As in any category, a subobject (on the vector space ) of a representation (on the vector space ) is a monomorphism . This natural transformation is specified by a single linear map . It’s straightforward to show that if is to be left-cancellable as an intertwinor, it must be left-cancellable as a linear map. That is, it must be an injective linear transformation from to .
Thus we can identify with its image subspace . Even better, the naturality condition means that we can identify the action of on with the restriction of to this subspace. The result is that we can define a subrepresentation of as a subspace of so that actually sends into itself for every . That is, it’s a subspace which is fixed by the action .
If is a subrepresentation of , then we can put the structure of a representation on the quotient space . Indeed, note that any vector in the quotient space is the coset of a vector . We define the quotient action using the action on : . But what if is another representative of the same coset? Then we calculate:
because sends the subspace back to itself.